\section{Why Does Insurance Work?} 
\label{sec:WhyInsuranceWorks} 

The Central Limit Theorem (CLT) \citep{Fel68, HoggCraig, HoggLossDist1984} explains that insurance works because large insurers issue many policies. As portfolio sizes increase the year to year, or portfolio to portfolio, variation in average claim costs approaches approaches zero and the insurer's average cost approaches the average cost for the population. Although individual policyholders' and small insurers' costs are very unpredictable, large insurers costs are very predictable. This means that while small insurers tend to earn excessive profits, or incur excessive losses, almost every year, large insurers earn high profits almost every year. This exposure to high and low losses is the hallmark of insurer inefficiency. Inefficient insurers, especially insurance risk assuming health care providers, are more likely to struggle financially and more likely to become insolvent than large insurers, as has happened for the last forty years \citep{Mayes2005}.

%As a simple example, suppose the average loss ratio for a population is 0.7500, and insurers devote 15\% of their premiums to no claim expenses. A very large insurer may have a loss ratio that falls between 0.7300 and 0.7700 in 99 out of 100 years. A much smaller insurer, insuring 1\% as many policyholders, will have a loss ratio that varies between 0.5500 and 0.9500 in 99 out of 100 years. The large insurer can meet the highest level of losses it is likely to incur (0.7700), out of currently available premiums (85\% of premium not devoted to non-loss expenses). The smallest insurer would need significant amounts of cash, before issuing policies, to be able to meet its obligations to policyholders and claimants at a loss ratio of 0.9500, which would result in an operating loss of 10\% of revenues. 

Despite the greater efficiency of large insurers, most health care (finance) reform recommendations ignore the reason insurance works, the Central Limit Theorem. Capitation advocates and other health care reformers suggest the opposite effect, that increased competition, among many small insurers will increase health care (finance) system efficiency and consumer benefits while reducing insurance costs. This is demonstrably false. Limited competition among very few, very large, and very efficient insurers is more efficient and produces higher policyholder benefits than large numbers of very small, very inefficient insurers.